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Abstract

Considerable attention is today devoted to the engineering of films widely used in
photocatalytic, solar energy converters, photochemical and photoelectrochemical cells,
dye-sensitized solar cells (DSSCs), to optimize electronic time response following
photogeneration. However, the precise nature of transport processes in these systems
has remained unresolved. To investigate such aspects of carrier dynamics, we have
suggested a model for the calculation of correlation functions, expressed as the Fourier
transform of the frequency-dependent complex conductivity σ(ω). Results are presented for the velocity correlation functions, the mean square deviation
of position and the diffusion coefficient in systems, like TiO2 and doped Si, of large interest in present devices. Fast diffusion occurs in short
time intervals of the order of few collision times. Consequences for efficiency of
this fast response are discussed in relation to nanostructured devices.

Keywords:

Nano Express

One of the most important aspects of nanostructures concerns charge transport, which
can be influenced by particle dimensions and assume different characteristics with
respect to those of bulk. In particular, if the mean free path of charges due to scattering
phenomena is larger than the particle dimensions, one has a mesoscopic system, in
which the transport depends on dimensions and one might correct the transport bulk
theories by considering this phenomenon. These problems occur also in a thin film,
in which the smallest dimension can be less than the free displacement and therefore
require variations to existing theoretical transport bulk models. This situation occurs
particularly in connection with metal oxide, like transparency, hardness, etc. Therefore
a rigorous knowledge of transport properties is to be acquired. To establish the applicability
limit of a bulk model and to investigate the time response of systems at nanoscale
we have performed a new approach based on correlation functions obtained by a Fourier
transform of the frequency-dependent complex conductivity of the system [1]. With this method it is possible to calculate these functions using experimental
data obtained by various films, like TiO2 and ZnO also in the form of nanowires, which have increasing interest for their technological,
chemical and biomedical applications and which are engineered to reach the desired
technical features. Also, the mesoporous films play a very important role, for their
applications in devices for energy generation, photocatalytic processes in environment
remediations and for the useful electronic propertiestechniques, in particular the
Time-resolved THz spectroscopy (TRTS) [2,3]. Starting from the Drude–Lorentz model [4,5] we have obtained directly the correlation function of velocities, the quadratic average
distance crossed by the charges as a function of time and the diffusion coefficient
D.

From a mathematical viewpoint, the Kubo relation of the linear response must be inverted.
But, due to the presence of a half Fourier-transform, it is necessary to modify this
relation in such a way that the whole time axis (-∞, +∞) occurs. This procedure is
not trivial and not previously found in the literature.

This new formula can be obtained by relying on linear response theory; we have started
considering a system with an hamiltonian of the form:

(1)

with H1 having small effects respect to H0, and negligible in the remote past (adiabatic representation). In the case of an
electric field of frequency ω we have:

(2)

For an electric field constant in space and depending on time as:

(3)

the time dependent corresponding current is:

(4)

Following the standard time-dependent approach [4], we derived a general formula for the linear response of a dipole moment density
in the β direction with the electric field directed in the α direction, where V is the volume of the system. This permits to deduce the susceptibility χ(ω), which is correlated to σ(ω) via the relation:

(5)

From Eq. 5 we have deduced the real part of σ(ω), denoted in the form σ'(ω) as:

(6)

where Sβα(ω)is the quantity:

(7)

The quantity 〈···〉T is the thermal average, and the exponential factor arises from equilibrium thermal
weights for Fermi particles. By considering the identity , Eq. 6 can be written in a form containing the velocity correlation function instead
of the position correlation function. Assuming the high temperature limit ħω < < KT as usual in systems to be considered in this paper, we obtain:

(8)

The integral in Eq. 8 spans the entire t axis, so we can perform the complete inverse Fourier transform of this equation.
It gives:

(9)

with V the volume of the system, K the Boltzmann's constant, T the temperature and σ'(ω) the real part of σ(ω), given by:

(10)

where n is the carrier density, ω0 the proper oscillator frequency, 1/τ the collision frequency [4,5].

The mean squared displacement in relation with the correlation function of velocities
is given by:

(11)

By integration of Eq. 9 with Eq. 10, we deduced all the results for , R2(t) and with x = t/τ [6].

The main advantage of this new formulation is the disposal of exact results for describing
the dynamic behaviour, as extracted by time-resolved techniques. In our analytical
procedure we have distinguished the case ωo = 0 from the case ωo ≠ 0. For this latter, three cases occur in connection with the sign of the quantity
. After obtaining the respective σ'(ω), we have found the poles of these functions and then the residues for integration
in the complex ω-plane via Cauchy theorem.

We have used our results for discussing transport in a conventional semiconductor
such as doped Si, non conventional TiO2 and other systems where anomalous transport has been found.

The most important characteristics of the results are illustrated by concrete examples
in Figures 1, 2, 3, 4, and 5.

In Figure 1, we show R2 for doped Si. For this semiconductor, the conductivity is the contribution of two
terms, a Drude–Lorentz term and a Drude term [7]. At large times the Drude–Lorentz term leads to an R2 approaching a constant value (see Figure 2), while the Drude term alone (Figure 1) is the dominant term at large times. Therefore for sufficiently large times, only
the Drude term survives.

We observe that the linear relation at large times becomes quadratic at smaller times.
The cross-over between the two regimes occurs at times comparable to the scattering
time. This means that diffusion occurs after sufficient time has elapsed so that scattering
events become significant, while at smaller times the motion is essentially ballistic.

In Figures 2 and 3R2 saturates at high t. The plateau value may assume high values so that R may be larger than the size of the nanoparticles composing the films. In general,
these features indicate quite enhanced mobility of carriers in the nanoporous films
at small times, in contrast with a commonly expected low mobility in a disordered
network.

From these figures we can evaluate the diffusion coefficient . It is remarkable that high D are obtained at t/τ of order unity. As an example, from Figure 3 the deduced that diffusion coefficient is D ~ 1 cm2/s for τ = 10-13s, i.e. comparable to the value ~1 cm2/s of the single crystal rutile [2].

From the other hand, much smaller D can develop at long times, with values D = 10-4–10-6 cm2/s typical of a disordered strong scattering system. So, our results indicate quite
different behaviour in Si where normal diffusion occurs, and in TiO2 where the Drude–Lorentz model indicates anomalous diffusion.

The physical reason and mechanism of such increase can be traced back to ballistic-like
motion of the carriers at early time when scattering is moderate yielding normal diffusion
satisfying Eintein's rule and to strong localization due to the scattering at long
times with anomalous diffusion with depression of D.

Figures 4 and 5 report the behaviour of the velocity correlation function.

We observe that, according to the equations of our model [8,9], the correlation function of velocities is never a single decreasing exponential
of time, but it is in general a more complicated combination of exponentials, or an
oscillating function of time.

When , there is a change of sign of velocity with respect to initial velocity, a backscattering
mechanism as indicated by Smith [10]: there are two regimes in the temporal response characterized by two different characteristic
times, the inverted region being dominated by the longer decay time, and the positive
velocity region being due to the shorter time; this region becomes the normal state
diffusion region when ω0 = 0. These two regimes will give rise to small and large diffusion constants respectively,
which will be discussed in connection with time-resolved techniques.

When , we observe the presence of damped oscillations of the velocity in time with strong
coupling leading to oscillating currents which average to zero in a sufficiently long
interval along with the diffusion constant. This regime appearing at large frequencies
for a given time constant does not seem to have been observed in real systems.

The results above give a precise indication on response times of a system subjected
to charge motion. In the case of doped Si [7] we have verified the Einstein rule, giving rise to normal diffusion. In the case
of TiO2 [2,3], anomalous diffusion is found with time-dependent diffusion coefficient vanishing
at long times and oscillating behaviour in time of the transport parameters. We have
compared our effective diffusion coefficients directly with experimental results [11-13] and with Monte Carlo simulations [14,15], which take into account the overall mechanisms of scattering, including phonons,
imperfections and doping centres, and traps, finding that the diffusion coefficient
reproduces the values of experimental or simulated coefficients.

We suggest the possibility that our results can give an explanation of the ultra-short
times and of high mobilities with which the charges spread in mesoporous nanoparticle
TiO2 systems, of deep interest in photocatalitic and photovoltaic systems [16,17]. In particular, the relative short times (few τ) with which charges can reach much larger distances than typical dimensions of nanoparticles
indicate easy diffusion for charges photoproduced inside the nanoparticles towards
the surface. The unexplained fact found experimentally of ultrashort injection of
charge carriers (particularly in Graetzel's cells) can be related to this phenomenon
[16,17].

Similar high diffusivity is found in a number of other devices, i.e. GaAs nanowires
and ZnO nanoparticles on which terahertz time-resolved spectroscopy has revealed different
time transport regimes with high diffusion processes at short times of the order of
the scattering time and longer time localized motion due to the effects of scattering
[18,19]. Interpretations of these results in terms of the model suggested here can be given.

Recently, an approach for converting nanoscale mechanical energy into electrical energy
has been suggested by using piezoelectric zinc oxide (ZnO) nanowires and TiO2 [20]. Such devices have been shown to convert mechanical energy into electric energy with
typical ∼1 nW output power per cm2 area. These unexpected efficiencies can be explained by anomalous high diffusion
in the oxides of the type presented here.

In summary, we have evaluated the correlation functions for systems for which the
Drude–Lorentz model is valid through the formulation of a new Drude–Lorentz-like model
[8,9], in which such functions can be obtained as complete Fourier transform of the real
part of the frequency-dependent complex conductivity σ(ω). From our results we deduce some important consequences connected with the nanometric
film systems, in particular the possibility of a fast response of the transport of
charge carriers with a direct consequence for the efficiencies of present devices
based on such systems. Of particular interest for nanostructures is the fact that
the limiting value of reaches several nanometers in only few τ times, which means that R becomes comparable to dimensions of nanoparticles in few scattering events. This
implies the possibility of having high mobility of carriers from and towards the surface
of nanostructures. This result has possible and interesting implications in photocatalysis
and in energy generators, i.e. in photochemical, photoelectrochemical cells and dye-sensitized
solar cells (DSSCs) [3]. The principal consequence is the possibility to have high charge conversion efficiencies
in particular time intervals. We can thus explain the rather unwaited experimental
result that some film oxides as TiO2, in which the percolative layer structure would be expected to provide a low mobility,
are in reality endowed with high response times of charge injection and with high
mobility.